FFTW 3.3.8: Real-to-Real Transform Kinds

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4.3.6 Real-to-Real Transform Kinds

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FFTW currently supports 11 different r2r transform kinds, specified by Chris@82: one of the constants below. For the precise definitions of these Chris@82: transforms, see What FFTW Really Computes. For a more colloquial Chris@82: introduction to these transform kinds, see More DFTs of Real Data. Chris@82:

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For dimension of size n, there is a corresponding “logical” Chris@82: dimension N that determines the normalization (and the optimal Chris@82: factorization); the formula for N is given for each kind below. Chris@82: Also, with each transform kind is listed its corrsponding inverse Chris@82: transform. FFTW computes unnormalized transforms: a transform followed Chris@82: by its inverse will result in the original data multiplied by N Chris@82: (or the product of the N’s for each dimension, in Chris@82: multi-dimensions). Chris@82: Chris@82:

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Chris@82: FFTW_R2HC computes a real-input DFT with output in Chris@82: “halfcomplex” format, i.e. real and imaginary parts for a transform of Chris@82: size n stored as: Chris@82:
Chris@82: r₀, r₁, r₂, ..., r_n/2, i_(n+1)/2-1, ..., i₂, i₁ Chris@82:
Chris@82: (Logical N=n, inverse is FFTW_HC2R.) Chris@82: Chris@82:
Chris@82: FFTW_HC2R computes the reverse of FFTW_R2HC, above. Chris@82: (Logical N=n, inverse is FFTW_R2HC.) Chris@82: Chris@82:
Chris@82: FFTW_DHT computes a discrete Hartley transform. Chris@82: (Logical N=n, inverse is FFTW_DHT.) Chris@82: Chris@82: Chris@82:
Chris@82: FFTW_REDFT00 computes an REDFT00 transform, i.e. a DCT-I. Chris@82: (Logical N=2*(n-1), inverse is FFTW_REDFT00.) Chris@82: Chris@82: Chris@82: Chris@82:
Chris@82: FFTW_REDFT10 computes an REDFT10 transform, i.e. a DCT-II (sometimes called “the” DCT). Chris@82: (Logical N=2*n, inverse is FFTW_REDFT01.) Chris@82: Chris@82:
Chris@82: FFTW_REDFT01 computes an REDFT01 transform, i.e. a DCT-III (sometimes called “the” IDCT, being the inverse of DCT-II). Chris@82: (Logical N=2*n, inverse is FFTW_REDFT=10.) Chris@82: Chris@82: Chris@82:
Chris@82: FFTW_REDFT11 computes an REDFT11 transform, i.e. a DCT-IV. Chris@82: (Logical N=2*n, inverse is FFTW_REDFT11.) Chris@82: Chris@82:
Chris@82: FFTW_RODFT00 computes an RODFT00 transform, i.e. a DST-I. Chris@82: (Logical N=2*(n+1), inverse is FFTW_RODFT00.) Chris@82: Chris@82: Chris@82: Chris@82:
Chris@82: FFTW_RODFT10 computes an RODFT10 transform, i.e. a DST-II. Chris@82: (Logical N=2*n, inverse is FFTW_RODFT01.) Chris@82: Chris@82:
Chris@82: FFTW_RODFT01 computes an RODFT01 transform, i.e. a DST-III. Chris@82: (Logical N=2*n, inverse is FFTW_RODFT=10.) Chris@82: Chris@82:
Chris@82: FFTW_RODFT11 computes an RODFT11 transform, i.e. a DST-IV. Chris@82: (Logical N=2*n, inverse is FFTW_RODFT11.) Chris@82: Chris@82:

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